1. Formal Languages, Automata and Models of Computation
CD5560
FABER
Formal Languages, Automata and Models of Computation
Lecture 7
Mälardalen University
2005

2. Content Context-Free Languages Push-Down Automata, PDA NPDA: Non-Deterministic PDA Formal Definitions for NPDAs NPDAs Accept Context-Free Languages Converting NPDA to Context-Free Grammar

3. Non-regular languages
Context-Free Languages
Regular Languages

4. Context-Free Languages

5. Context-Free Languages
Grammars
Pushdown
Automata
stack
automaton
(CF grammars are
defined as generalized Regular Grammars)

6. Definition: Context-Free Grammars
Variables
Terminal
symbols
Start
variables
Productions of the form:
is string of variables and terminals

7. Definition: Regular Grammars
Variables
Terminal
symbols
Start
variables
Right or Left Linear Grammars. Productions of the form: or
is string of terminals

8. Pushdown Automata PDAs

9. Pushdown Automaton - PDA
Input String
Stack
States

10. A PDA can write symbols on stack and read them later on.
The Stack
A PDA can write symbols on stack and read them later on.
POP reading symbol PUSH writing symbol
All access to the stack only on the top!
(Stack top is written leftmost in the string, e.g. yxz)
A stack is valuable as it can hold an unlimited amount of information.
The stack allows pushdown automata to recognize some non-regular languages.

11. The States Pop old reading Push new Input stack symbol writing symbol

12. input stack top Replace
(An alternative is to start and finish with empty stack)

13. input
Push
top
stack

14. input
Pop
top
stack

15. input
No Change
top
stack

16. Example 3.7 Salling: Input Time 0
A PDA for simple nested parenthesis strings
Time 0
Input
Stack

17. Example 3.7
Time 1
Input
Stack

18. Example 3.7
Time 2
Input
Stack

19. Example 3.7
Time 3
Input
Stack

20. Example 3.7
Time 4
Input
Stack

21. Example 3.7
Time 5
Input
Stack

22. Example 3.7
Time 6
Input
Stack

23. Example 3.7
Time 7
Input
Stack

24. NPDAs Non-deterministic Push-Down Automata

25. Non-Determinism

26. A string is accepted if:
All the input is consumed
The last state is a final state
Stack is in the initial condition
(either: empty (when we started with empty stack),
or: bottom symbol reached, or similar)

27. Example NPDA
is the language accepted by the NPDA:

28. Example NPDA
NPDA
(Even-length palindromes)

29. Pushing Strings
Pop
symbol
Input
symbol
Push
string

30. Example:
input
pushed
string
stack
top
Push

31. Another NPDA example
NPDA

32. Execution Example:
Time 0
Input
Stack
Current state

33. Time 1
Input
Stack

34. Time 2
Input
Stack

35. Time 3
Input
Stack

36. Time 4
Input
Stack

37. Time 5
Input
Stack

38. Time 6
Input
Stack

39. Time 7
Input
Stack
accept

40. Formal Definitions for NPDAs

41. Transition function:

42. Transition function: new state current state current stack top
new stack top
current input symbol
An unspecified transition function is to the null set and represents a dead configuration for the NPDA.

43. Formal Definition Non-Deterministic Pushdown Automaton NPDA Final
States
Input
alphabet
Stack
Transition
function
Final
states
start
symbol

44. Instantaneous Description
Current
state
Current
stack
contents
Remaining
input

45. Instantaneous Description
Example
Instantaneous Description
Input
Time 4:
Stack

46. Instantaneous Description
Example
Instantaneous Description
Input
Time 5:
Stack

47. We write
Time 4
Time 5

48. A computation

49. A computation

50. A computation

51. A computation

52. A computation

53. A computation

54. A computation

55. A computation

56. For convenience we write

57. Formal Definition
Language of NPDA
Initial state
Final state

58. Example:
NPDA :

59. NPDA :

60. Therefore:
NPDA :

61. NPDAs Accept Context-Free Languages

62. Theorem
Context-Free
Languages
(Grammars)
Accepted by
NPDAs

63. Proof - Step 1:
Context-Free
Languages
(Grammars)
Languages
Accepted by
NPDAs
Convert any context-free grammar
to a NPDA with

64. Proof - Step 2:
Context-Free
Languages
(Grammars)
Languages
Accepted by
NPDAs
Convert any NPDA to a context-free
grammar with:

65. Converting Context-Free Grammars to NPDAs

66. An example grammar:
What is the equivalent NPDA?

67. Grammar
NPDA

68. The NPDA simulates
leftmost derivations of the grammar
L(Grammar) = L(NPDA)

69. Grammar:
A leftmost derivation:

70. NPDA execution:
Time 0
Input
Stack
Start

71. Time 1
Input
Stack

72. Time 2
Input
Stack

73. Time 3
Input
Stack

74. Time 4
Input
Stack

75. Time 5
Input
Stack

76. Time 6
Input
Stack

77. Time 7
Input
Stack

78. Time 8
Input
Stack

79. Time 9
Input
Stack

80. Time 10
Input
Stack
accept

81. In general
Given any grammar
We can construct a NPDA
With

82. Constructing NPDA from grammar
Top-down parser
For any production
For any terminal

83. Grammar generates string
if and only if
NPDA accepts

84. Therefore:
For any context-free language
there is an NPDA
that accepts the same language

85. Which means
Context-Free
Languages
(Grammars)
Accepted by
NPDAs

86. Converting NPDAs to Context-Free Grammars

87. For any NPDA
we will construct
a context-free grammar with

88. The grammar simulates the machine
A derivation in Grammar
in NPDA
Input processed
Stack contents
terminals
variables

89. First we modify the NPDA so that It has a single final state
Some Simplifications
First we modify the NPDA so that
It has a single final state
It empties the stack
when it accepts the input
Original NPDA
Empty Stack

90. Second we modify the NPDA transitions. All transitions will have form:
which means that each move
increases/decreases stack by a single symbol.

91. Those simplifications do not affect generality of our argument.
It can be shown that for any NPDA there exists an equivalent one having above two properties
i.e.
the equivalent NPDA with a single final state which empties its stack when it accepts the input, and which for each move increases/decreases stack by a single symbol.

92. Example of a NPDA in an appropriate form

93. The Grammar Construction
In grammar
Stack symbol
Variables:
states
Terminals:
Input symbols of NPDA

94. For each transition:
we add production:

95. For each transition:
we add production:
for all states

96. Stack bottom symbol
Start Variable
Start state
(Single) Final state

97. Example
Grammar production:

98. Grammar productions:

99. Grammar production:

100. Resulting Grammar

101. Resulting Grammar, cont.

102. Resulting Grammar, cont.

103. Derivation of string

104. In general, in grammar:
if and only if
is accepted by the NPDA

105. Explanation
By construction of Grammar:
if and only if
in the NPDA going from to
the stack doesn’t change below
and is removed from stack

106. We have shown the procedure to convert
any NPDA to a context-free
grammar with:
which means
Context-Free
Languages
(Grammars)
Accepted by
NPDAs

107. Therefore Context-Free Languages Languages Accepted by (Grammars)
NPDAs
END OF PROOF

108. Example (Sudkamp 8.1.2)
Language consisting solely of a’s or an equal number of a´s and b´s.

109. When scanning an a in state q0 two transitions are possible
When scanning an a in state q0 two transitions are possible. A string of the form aibi is accepted by a computation in states q0 and q1.